I don't think there really is an answer to the specific who. The how is more important: you must ensure that pointing out the fallacies does not hinder the debate, i.e. that it continues.
Two pitfalls here are:
Pointing out logical fallacies does not invalidate other arguments ("You are using a logical fallacy, so now you have completely lost the argument"): The fallacy fallacy
Especially when the opposing debater points out a fallacy, the discussion may evolve into arguing about the fallacy itself, thereby forcing the opposing debater into the defense. That is a place (s)he wants to avoid!
If you want to avoid those effects, or if you want to distinguish fallacies ASAP (during the debate) that would favor having the moderator point them out:
- "Isn't that a strawman fallacy? I don't have the impression that that is what your opponent is saying."
- "Please do not bring it as the false dichotomy XXXX. This is not zero-sum game where YYYY."
- "You are stepping out of line now. This is an ad hominem attack"
OTOH You could argue that the debaters should be able to address these pitfalls themselves ;-)
If there is more time available and you have an active community that follows up the debate, you could leave that up to them in the comments.
I recently started following the You Are Not So Smart podcast who are running a series about logical fallacies (and they have three guests that you may want to visit/follow as well). For each fallacy type they specifically address that second issue: when you spot a fallacy used by your opponent, how do you address that without putting yourself in the defensive.